Help Needed: Evaluating a Limit

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In summary, a limit in mathematics is a way to analyze the behavior of a function as the input approaches a certain value without evaluating the function at that point. To evaluate a limit, one must analyze the function's values near the specified point and may use algebraic techniques. One-sided limits consider the function's behavior from one direction, while two-sided limits consider both directions. A limit can still exist if the function is not defined at the specified point. Limits and continuity are closely related, with a continuous function having its limit equal to its actual value at a specific point.
  • #1
johnnyICON
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Limits! help!

I have this question that I am not too sure of how to do, can anyone help me?

Evaluate: [itex]\lim_{x \to 0} \frac{\sin{x}}{\sqrt{x}}[/itex]
 
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  • #2
When x approaches 0, sin x approaches x. so you get
[tex]\frac{x}{\sqrt{x}}=\sqrt{x}[/tex]
 
  • #3
Or remember that [itex]\frac{sinx}{x}[/itex] goes to 1. Then write
[tex]\frac{\sin{x}}{\sqrt{x}}=\frac{\sin{x}}{x}\sqrt{x}[/tex]
 
  • #4
awesome, thanks! :smile:
 

1. What is a limit in mathematics?

A limit in mathematics is a fundamental concept used in calculus to describe the behavior of a function as the input approaches a certain value. It is a way to analyze the values of a function near a specific point without having to evaluate the function at that point.

2. How do you evaluate a limit?

To evaluate a limit, you need to determine the behavior of the function near the specific point. This can be done by analyzing the values of the function as the input values get closer and closer to the specified point. You can also use algebraic techniques such as factoring or simplifying to help evaluate the limit.

3. What is the difference between a one-sided limit and a two-sided limit?

A one-sided limit only considers the behavior of the function as the input approaches the specified point from one direction, either the left or the right. A two-sided limit, on the other hand, looks at the behavior of the function as the input approaches the specified point from both the left and the right.

4. Can a limit exist if the function is not defined at the specified point?

Yes, a limit can still exist even if the function is not defined at the specified point. This can happen if the function has a hole or a jump in the graph at that point. In this case, the limit would be the value that the function approaches as the input gets closer and closer to the specified point.

5. How do limits relate to continuity?

Limits and continuity are closely related concepts in mathematics. A function is considered continuous if its limit at a specific point is equal to its actual value at that point. This means that the function is smooth and has no breaks or jumps in its graph at that point.

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